A $C^1$ Map into a Euclidean Space is Differentiable at Every Point

Let $n$ and $p$ be natural numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $f=(f_1,\dots,f_p):U\to\mathbb{R}^{p}$ with coordinate functions $f_k:U\to\mathbb{R}$, and let $a\in U$. Suppose that $f$ is of class $C^{1}$ on $U$, in the sense of clause 1 of that definition.

Then the Jacobian matrix $Df(a)$ is defined, and $f$ is differentiable at $a$ with derivative matrix $Df(a)$.